Computationally, there are two general ways to treat problems with complex geometry. One way is to use unstructured grids. The other is to use overset grids. Overset grids are formed by overlapping structured grids. In this chapter, the basic idea of overset grids methodology and its implementation are discussed.

Basic Concept of Overset Grids

To illustrate the basic idea of overset grids, consider the problem of computing the scattering of acoustic waves by a solid cylinder in two dimensions. In the space around the cylinder, the coordinates of choice for computing the solution is the cylindrical polar coordinates centered at the axis of the cylinder. This coordinate system provides a set of body-fitted coordinates and, hence, a body-fitted mesh when discretized around the cylinder. One significant advantage of using a body-fitted grid is the relative ease in enforcing the no-through-flow wall boundary condition using the ghost point method or other methods. Away from the cylinder, acoustic waves propagate with no preferred direction. The natural coordinate system to use is the Cartesian coordinates. Therefore, to take into account the advantages stated, one may use a polar mesh around the cylinder and a Cartesian mesh away from the cylinder with an overlapping mesh region. The overlapping mesh region is for data transfer from one set of grids to the other and vice versa.

Many aeroacoustics problems involve multiple length and time scales. This should not be difficult to understand. For, in addition to the intrinsic sizes and scales of the noise sources, the acoustic wavelength is an inherent length scale of the problem. In many instances, the length scale of the noise source differs greatly from the acoustic wavelength. This leads to a large disparity in length scales as in classical multiscales problems. For example, in supersonic jet noise, Mach wave radiation is generated by the instability waves of the jet flow. The instability waves are supported by the thin shear layer of the jet. In the region near the nozzle exit, the averaged shear layer thickness is about 0.1D, where D is the jet diameter. The acoustic wavelength, on the other hand, is two or more jet diameters long. Thus, there is an order of magnitude difference between those characteristic lengths. In sound scattering problems, the length scale of the surface geometry of the scatterers may be much smaller than the acoustic wavelength. This occurs very often in edge scattering and diffraction problems. A concrete example is the radiation of fan noise from a jet engine inlet. The acoustic wavelength could be much longer than the radius of the lip of the engine inlet. To obtain an accurate numerical solution of the inlet diffraction problem, a fine mesh is needed around the lip region. Oftentimes, an aeroacoustics problem becomes a multiscales problem because of the change in the physics governing the different parts of the computational domain. An example is the shedding of vortices at the edge of a resonator or a sharp edge of a solid body induced by high-intensity incident sound waves. Away from the solid surface, the fluid is nearly inviscid, but close to the wall, the viscosity effect dominates. The oscillatory motion of the incident sound waves induces a very thin Stokes layer on the solid surface. The Stokes layer rolls up at the corner of a solid surface to form vortices that shed periodically. To simulate the vortex shedding process, therefore, it is necessary to use very fine mesh close to the solid surface and around the corner to resolve the Stokes layer. But away from the solid surface, a coarse mesh with 7 mesh points per acoustic wavelength is all that is needed to capture the sound waves accurately using the 7-point stencil dispersion-relation-preserving (DRP) scheme.

Acoustics are governed by the compressible Navier-Stokes equations. In most cases, molecular viscosity is unimportant, so that the use of Euler equations is sufficient. In solving the Navier-Stokes or Euler equations computationally, the first step is to perform a discretized approximation to the spatial and temporal derivatives. This converts the partial differential equations to a set of partial difference equations. However, one must recognize that the solutions of the discretized equations are not the same as those of the original partial differential equations. A central effort of computational aeroacoustics (CAA) is to understand mathematically the behavior of the solution of the discretized equations and to quantify and minimize the error. Here, error is referred to as the difference between the solution of the original partial differential equations and the partial difference system.

Invariably, finite difference approximation of the governing equations of acoustics will result in a dispersive wave system (Vichnevetsky and Bowles, 1982; Trefethen, 1982; Tam and Webb, 1993; Tam 1995), even though the waves supported by the original partial differential equations are nondispersive. This is an extremely important point and should be clearly understood by all CAA investigators and users.

The objective of this chapter is to discuss how to design a computation code to simulate an aeroacoustic phenomenon. In previous chapters, many methods and elements of numerical computation were discussed. In this chapter, they are to be synthesized to form a computer simulation code. The basic elements/ingredients of a good simulation algorithm in computational aeroacoustics would consist of the following:

1. A computational model containing all essential physics of the aeroacoustic phenomenon

2. A properly chosen computational domain

3. A well-designed computational grid

4. A least dispersive and dissipative high-resolution time marching algorithm

5. A time step that ensures numerical stability and good resolution

6. A set of high-quality boundary conditions for both exterior and interior boundaries

7. A properly chosen distribution of artificial selective damping to suppress the generation and propagation of spurious short waves

8. A set of properly prescribed initial conditions

Basic Elements of a CAA Code

Each of the eight elements in this list affects the simulation in some way. Some exert an influence on numerical stability. Some affect the accuracy and quality of the computed solution. Others control and influence the computation time. A more detailed consideration of some of these elements in this list is provided next.

A computational domain is inevitably finite. For open domain problems, this automatically creates a set of artificial exterior boundaries. There are two basic reasons why exterior boundary conditions are needed at the artificial boundaries. First, exterior boundary conditions must reproduce the effects the outside world exerts on the flow inside the computation domain. Since only the computed results inside the computational domain are known, in general, it is not possible to know the outside influence. For this reason, a computation domain is often taken as large as possible so that all sources are inside. This will minimize any external influence that is unaccounted for.

Another reason for imposing exterior boundary conditions is to avoid the reflection of outgoing disturbances back into the computational domain and thus contaminate the computed solution. One way to avoid reflection is to construct boundary conditions in such a way to allow the smooth exit of all disturbances.

In this chapter, it will be assumed that there is little external influence and there are no incoming disturbances. Issues of external influence and incoming entropy, vorticity, or acoustic waves will be considered in later chapters.

As discussed previously, the linearized Euler equations support three types of waves. Two types of these waves, namely, the entropy and vorticity waves, are convected downstream by the mean flow. The acoustic waves, however, propagate and radiate out in all directions if the mean flow is subsonic. Thus, at a subsonic inflow region, the outgoing waves consist of acoustic waves alone. In the outflow region, the outgoing disturbances now comprise of all three types of waves. As a result, it is prudent to develop separate inflow and outflow boundary conditions.

High-quality boundary conditions are an essential part of computational aeroacoustics (CAA). Because a computational domain is necessarily finite in size, numerical boundary conditions play several diverse roles in numerical simulation. First and foremost, they must assist any outgoing disturbances to leave the computational domain with little or no reflection. The alternative is to use a perfectly absorbing layer as a numerical boundary treatment. Such a layer absorbs all outgoing disturbances without reflection as in the case of an anechoic chamber. In addition, if the problem to be simulated involves incoming disturbances, then these disturbances must be generated by the boundary conditions prescribed at the outer boundary of the computational domain. Furthermore, if there are flows that are originated from outside the computational domain, they must be reproduced by the boundary treatment as well. In this chapter, methods to construct numerical boundary conditions that perform these various functions are discussed.

Boundaries with Incoming Disturbances

In many aeroacoustics problems, there are disturbances that enter the computational domain through the outer boundary. For example, in computing the scattering characteristics of an object, acoustic waves must be allowed to pass through the boundary of a computational domain in a specified direction and intensity. The scattering phenomenon produces scattered waves. These waves are radiated in all directions. They propagate to the far field as outgoing waves through the boundary of the computational domain. In this case, the boundary condition of the computational domain must take on the responsibility of generating the incoming sound and, at the same time, they also serve as radiation boundary conditions for the scattered waves. To handle the dual role, a split-variable method has been developed. The essence of this method is discussed below.